CN103117969A - Multi-modulus blind equalization method using wavelet frequency domain transform based on fractional lower order statistics - Google Patents

Abstract

The present invention proposes a fractional low-order statistic modulus transform wavelet frequency-domain multi-mode blind equalization method. The method utilizes fractional low-order statistic to suppress α-stable distribution noise, and utilizes modulus transform to transform multiple amplitude moduli It is transformed into a single amplitude modulus, and then the orthogonal wavelet transform and fast Fourier transform are introduced into the multi-mode blind equalization method. The method realizes the effective equalization of the M-order quadrature amplitude modulation signal MQAM through the modulus transformation, and at the same time reduces the calculation amount by using the FFT and the overlap preservation method, and performs the orthogonal wavelet transformation on the signal before it enters the equalizer, The autocorrelation of the input signal is reduced. The method of the invention has faster convergence speed and smaller steady-state error, and has certain application value in the field of underwater acoustic communication.

Description

A multi-mode blind equalization method based on fractional low-order statistic modulus transform wavelet frequency domain

technical field

The invention belongs to the technical field of underwater acoustic wireless communication, and specifically refers to a fractional low-order statistic modulus transform wavelet frequency-domain multi-mode blind equalization method.

Background technique

In underwater acoustic communication systems, inter-symbol interference (ISI) caused by multipath effects and channel distortion reduces the transmission rate and reliability of information. In order to overcome the ISI in the underwater acoustic channel, blind adaptive equalization technology is needed to eliminate it. Blind adaptive equalization technology has become a research hotspot in the field of underwater acoustic communication because it does not need to transmit training sequences. The time-domain constant modulus algorithm (CMA) is the most widely used, which is very suitable for the equalization of constant modulus signals. However, the constellation of the M-order quadrature amplitude modulation signal (MQAM) does not match its statistical modulus, resulting in a poor equalization effect of CMA on MQAM (such as 16QAM signal). After the multiple amplitude modulus values of the non-modulus MQAM are transformed into a single amplitude modulus value through the modulus transformation, it is equalized, which effectively reduces the steady-state error.

In the traditional blind adaptive equalization method (referred to as the blind equalization method), the channel noise is assumed to be Gaussian noise, but a large number of studies in recent years have shown that some noise in the actual channel often shows strong impulsiveness and is not Fully obey the Gaussian distribution model, but a generalized Gaussian distribution model called α-stable distribution. In α-stable distributed noise, the second-order statistics of the signal do not exist, and the signal processing method based on the second-order statistics of the signal is no longer applicable, so it is necessary to use fractional low-order statistics to analyze and process the signal.

The frequency domain constant modulus algorithm (FCMA) transforms the traditional blind equalization method in the time domain into the frequency domain for blind equalization processing. (overlap-save law, OSL), which reduces the calculation amount of the time-domain blind equalization method. The orthogonal wavelet transform is introduced into the blind equalization method, and the good decorrelation of the equalizer input signal by the orthogonal wavelet is used to accelerate the convergence speed.

To sum up, in the prior art, there is no complete and effective technical solution for the problem of multi-mode blind equalization under the condition of α-stable distributed noise channel.

Contents of the invention

The technical problem to be solved by the present invention is to overcome the deficiencies of the prior art, aiming at the performance degradation of the time-domain constant modulus method CMA in α-stable distributed noise and its inability to effectively balance the 16-order quadrature amplitude modulation multi-mode signal, a method is proposed A fractional low-order statistic modulus transform wavelet frequency-domain multi-mode blind equalization method WT-FLOSMTFMMA; the method uses fractional low-order statistic to suppress α-stable distribution noise, and uses modulus transform to transform multiple amplitude moduli into Single amplitude modulus to reduce steady-state error; use fast Fourier transform and overlap preservation method to reduce the calculation amount of traditional blind equalization method; use orthogonal wavelet transform to reduce the autocorrelation of equalizer input signal.

In order to solve the problems of the technologies described above, the technical solution adopted in the present invention is:

A fractional low-order statistic modulus transform wavelet frequency domain multi-mode blind equalization method, comprising the following steps:

Step A, pass the transmitted signal a(n) through the impulse response channel c(n) to obtain the channel output vector b(n), where n is a time series;

Step B, using α-stable distributed noise w(n) and the channel output vector b(n) described in step A to obtain the equalizer input signal y(n) before orthogonal wavelet transform: y(n)=w(n) +b(n);

Step C, get the real part y _r (n) and the imaginary part y i (n) to the equalizer input signal y(n) described in step B, then for the real part y _r ( _n ) and the imaginary part y _i (n ) respectively carry out orthogonal wavelet transform, then the signal after orthogonal wavelet transform is

v _r (n)=Qy _r (n), v _i (n)=Qy _i (n)

where Q is the orthogonal wavelet transform matrix, v _r (n) and v _i (n) are the real part y _r (n) and imaginary part y _i (n) of the input signal y(n) of the time domain equalizer respectively The signal components after orthogonal wavelet transform, the real part Z _r (N) and the imaginary part Z _i (N) of the frequency domain equalizer output Z (N) are respectively

Z _r (N)=R _r (N)F _r (N), Z _i (N)=R _i (N)F _i (N)

In the formula, N represents the number of data blocks with length L, L is the length of equalizer weight vector, F _r (N) and F _i (N) are the real part and The imaginary part, R _r (N) and R _i (N) are the frequency domain real part and imaginary part of v _r (n) and v _i (n) after fast Fourier transform;

Step D, perform inverse Fourier transform on the real part Z _r (N) and the imaginary part Z _i (N) of the frequency domain equalizer output signal Z(N) in step C respectively to obtain the time domain equalizer output signal z(n) The real part z _r (n) and the imaginary part z _i (n) of .

In the step C, the calculation steps of the frequency domain equalizer weight vector F(N) are as follows:

Step C-1, calculate the real part e _rt (n) and the imaginary part e _it (n) of the time domain error function e _t (n) of the modulus transformation, the calculation formula is as follows:

${\mathrm{<span\; class="notranslate">\; e</span>}}_{\mathrm{<span\; class="notranslate">\; rt</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; rt</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; rt</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; p</span>}}<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; e</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; p</span>}}$

${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; rt</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; \{</span>{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; rt</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; p</span>}}<span\; class="notranslate">\; \}</span>}{\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; \{</span>{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; rt</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; p</span>}}<span\; class="notranslate">\; \}</span>}<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; =</span>\frac{\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; \{</span>{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; p</span>}}<span\; class="notranslate">\; \}</span>}{\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; \{</span>{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; a</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; p</span>}}<span\; class="notranslate">\; \}</span>}$

与

分别为a_rt(n)与a_it(n)的p阶统计模值，z_rt(n)与z_it(n)分别为时域均衡器输出信号z(n)的实部z_r(n)与虚部z_i(n)经过模值变换后的实部与虚部；步骤C-2，对模值变换时域误差函数e_t(n)的实部e_rt(n)与虚部e_it(n)作傅里叶变换后，得到模值变换频域误差函数E_t(N)的实部E_rt(N)与虚部E_it(N)；In the formula, |·| is a modulo operation, p is a positive number greater than zero and less than 1, E{ } represents the mathematical expectation, a _rt (n) and a _it (n) are the transmitted signal a(n) The signal components of the real part a _r (n) and the imaginary part a _i (n) after the modulus transformation respectively,

and

are the p-order statistical moduli of a _rt (n) and a _it (n) respectively, and z _rt (n) and z _it (n) are the real part z _r (n) of the time-domain equalizer output signal z(n) respectively ) and imaginary part z _i (n) after modulus transformation; step C-2, the real part e _rt (n) and imaginary part of modulus transformed time domain error function e _t (n) After e _it (n) is Fourier transformed, the real part E _{rt (} N) and the imaginary part E rt (N) of the frequency domain error function E _t (N) of the modular value transformation are obtained;

Step C-3, calculating the frequency domain equalizer weight vector F(N), the formula of the iterative process is:

${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&mu;</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; rt</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>$

$<span\; class="notranslate">\; \&Center\; Dot;</span>\mathrm{<span\; class="notranslate">\; sign</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; rt</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; Z</span>}}_{\mathrm{<span\; class="notranslate">\; rt</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; sign</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; Z</span>}}_{\mathrm{<span\; class="notranslate">\; rt</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}^{<span\; class="notranslate">\; *</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span>$

${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&mu;</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>$

$<span\; class="notranslate">\; \&Center\; Dot;</span>\mathrm{<span\; class="notranslate">\; sign</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; Z</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; sign</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; Z</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{<span\; class="notranslate">\; *</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span>$

和

表示频域均衡器输入信号R(N)的实部R_r(N)与虚部R_i(N)的共轭；

为

的快速傅里叶变换，且

的获取公式为：In the formula, μ is the iterative step size, sign( ) represents the sign operation, Z _rt (N) and Z _it (N) are the real values of the output signal Z _t (N) of the frequency domain equalizer after modulus transformation respectively part and imaginary part;

and

Represents the conjugate of the real part R _r (N) and the imaginary part R _i (N) of the frequency domain equalizer input signal R (N);

for

The fast Fourier transform of , and

The acquisition formula is:

${\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; diag</span>}<span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span><span\; class="notranslate">\; \&CenterDot;</span><span\; class="notranslate">\; \&Center\; Dot;</span><span\; class="notranslate">\; \&CenterDot;</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1,0</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span><span\; class="notranslate">\; \&CenterDot;</span><span\; class="notranslate">\; \&Center\; Dot;</span><span\; class="notranslate">\; \&Center\; Dot;</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; J</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>$

与

分别表示对u_j，k(n)与

的平均功率估计，可由下式递推得到：Among them, diag[ ] represents a diagonal matrix,

and

Respectively for u _{j, k} (n) and

The average power estimate of can be recursively obtained by the following formula:

${\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}_{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}_{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; u</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}$

${\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}_{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}_{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}$

In the formula, u _{j, m} (n) is the wavelet transformation coefficient with scale parameter j and translation parameter m, s _{J, m} (n) is the scale transformation coefficient with scale parameter J and translation parameter m, and J is the wavelet The maximum scale of decomposition, k is the translation parameter of the corresponding wavelet function under the scale parameter j, k _J represents the maximum translation of the wavelet function under the maximum scale J, β _σ is the smoothing factor, and 0<β _σ <1.

The beneficial effects of the present invention are: the present invention proposes a fractional low-order statistic modulus transformation wavelet frequency domain multi-mode blind equalization method, the method utilizes the fractional low-order statistic to suppress α-stable distribution noise, The transformation transforms multiple amplitude moduli into a single amplitude modulus, and then introduces the orthogonal wavelet transform and fast Fourier transform into the multi-mode blind equalization method. The method realizes the effective equalization of the M-order quadrature amplitude modulation signal MQAM through the modulus transformation, and at the same time reduces the calculation amount by using the FFT and the overlap preservation method, and performs the orthogonal wavelet transformation on the signal before it enters the equalizer, The autocorrelation of the input signal is reduced. The method of the invention has faster convergence speed and smaller steady-state error, and has certain application value in the field of underwater acoustic communication.

Description of drawings

Fig. 1: It is the principle diagram of WT-FLOSMTFMMA of the method of the present invention.

Figure 2: It is a 16QAM signal constellation diagram.

Figure 3: is the schematic diagram of MTMMA.

Figure 4: is the schematic diagram of FCMA.

Figure 5: is the schematic diagram of MTFMMA.

Figure 6: is the schematic diagram of FLOSMTFMMA.

Figure 7: Implementation example result diagram, Figure 7(a) mean square error curves of the three methods, 7(b) FLOSFCMA output constellation diagram, 7(c) FLOSMTFMMA output constellation diagram, 7(d) WT-FLOSMTFMMA output constellation diagram .

Figure 8: is the result diagram of the implementation example under different frequency domain error function exponents, 8(a) mean square error curves of the three methods, 8(b) FLOSFCMA output constellation diagram, 8(c) FLOSMTFMMA output constellation diagram, 8 (d) WT-FLOSMTFMMA output constellation diagram.

Detailed ways

Below in conjunction with the accompanying drawings, a kind of fractional low-order statistic modulus transformation wavelet frequency domain multi-mode blind equalization method proposed by the present invention is described in detail:

A. Modulo value transformation method

的平方根，a(n)的虚部a_i(n)模值|a_i(n)|也不等于a_i(n)的二阶统计模值

的平方根；|·|为取模运算，E{·}表示求数学期望。现以16QAM信号为例说明，能通过模值变换方法使16QAM信号a(n)的幅度模值与其统计模值相等。The amplitude modulus |a(n)| of the M-order quadrature amplitude modulation signal (MQAM) a(n) is not a constant, nor equal to its second-order statistical modulus R ² =E{|a(n)|} ⁴ / E{|a(n)|} is the square root of ² ; the real part of a(n) a _r (n) modulus |a _r (n)| is not equal to the second-order statistical modulus of a _r (n)

The square root of the imaginary part of a(n) a _i (n) modulus |a _i (n)| is not equal to the second-order statistical modulus of a _i (n)

The square root of ; |·| is a modulo operation, and E{·} represents mathematical expectation. Taking the 16QAM signal as an example to illustrate, the amplitude modulus of the 16QAM signal a(n) can be made equal to its statistical modulus through the modulus transformation method.

(

表示开方运算),此时

因此，CMA方法不能对16QAM信号进行有效均衡。将发射信号模值变换方程定义为Figure 2 shows the constellation diagram of the 16QAM signal. It can be seen from Figure 2 that the 16QAM signal is a very modulo-complex signal a(n), so the real part and the imaginary part of the signal are considered separately. In the 16QAM constellation diagram, the amplitude modulus values corresponding to the real part a _r (n) and the imaginary part a _i (n) of the a(n) signal are respectively |a _r (n)|=1 or 3, |a _i ( n)|=1 or 3; the statistical modulus of the real part a _r (n) and the imaginary part a _i (n) of a(n) are respectively

(

represents the square root operation), at this time

Therefore, the CMA method cannot effectively equalize the 16QAM signal. The transmit signal modulus transformation equation is defined as

|a _rt (n)|=||a _r (n)|-2|,|a _it (n)|=||a _i (n)|2| (1)

此时，模值变换后发射信号的模值与其统计模值相等，即 $\left|a_{\mathrm{it}}\left(n\right)\right|=\sqrt{R_{\mathrm{it}}^2}=1.$ In the formula, |a _rt (n)| and |a _it (n)| are the modulus values of real part _{a rt (} n) and imaginary part a _it (n) of transmitted signal at (n) after modulus transformation respectively , Substituting |a _r (n)| and |a _i (n)| into formula (1) to get |a _rt (n)|=1, a _it (n)|=1, then the signal a is transmitted after modulus transformation The statistical modulus values of the real part and imaginary part of _t (n) are respectively

At this time, the modulus of the transmitted signal after modulus transformation is equal to its statistical modulus, that is $\left|a_{\mathrm{it}}\left(\mathrm{no}\right)\right|=\sqrt{R_{\mathrm{it}}^2}=1.$

B. Multi-mode Blind Equalization Method in Modulus Transformation Time Domain

After the multi-mode 16QAM signal is changed into a single mode value through the mode value conversion, it can be effectively equalized by the CMA method. This blind equalization method, which converts multi-mode signals into single-mode signals through modulus transformation, is called modulo-value transform time-domain multi-mode blind equalization method (MTMMA), as shown in FIG. 3 . The cost function of this method is defined as

${\mathrm{<span\; class="notranslate">\; J</span>}}_{\mathrm{<span\; class="notranslate">\; MTMMA</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; \{</span>{<span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; rt</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; rt</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ]</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; +</span><span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{<span\; class="notranslate">\; ]</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; )</span>$

In the formula, J _MTMMA is the cost function of MTMMA; |z _rt (n)|=||z _r (n)|-2|, |z _it (n)|=||z _i (n)|-2| are respectively the modulus values of the real part z _rt (n) and the imaginary part z _it (n) of the output signal z _t (n) of the modulus transform equalizer; z _r (n) and z _i (n) are the time domain equalization The real and imaginary parts of the output signal z(n) of the device.

The weight vector iteration formula of MTMMA is

$f_r(\mathrm{no}+1)=f_r\left(\mathrm{no}\right)+\mathrm{\&mu;}\left|z_{\mathrm{rt}}\left(\mathrm{no}\right)\right|[1-|z_{\mathrm{rt}}\left(\mathrm{no}\right){|}^2\left]\mathrm{sign}\right[z_r\left(\mathrm{no}\right)]{\mathrm{the\; y}}_r^{*}\left(\mathrm{no}\right)$ (3)

${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&mu;</span>}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ]</span>\mathrm{<span\; class="notranslate">\; sign</span>}<span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>{\mathrm{<span\; class="notranslate">\; the\; y</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{<span\; class="notranslate">\; *</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>$

与

分别为时域均衡器输入信号y(n)的实部y_r(n)与虚部y_i(n)的共轭，f_r(n)与f_r(n)分别为时域均衡器权向量f(n)的实部与虚部。In the formula, μ is the iterative step size, and sign(·) represents the sign operation,

and

are the conjugates of the real part y _r (n) and the imaginary part y _i (n) of the input signal y(n) of the time domain equalizer respectively, f _r (n) and f _r (n) are the weights of the time domain equalizer The real and imaginary parts of the vector f(n).

C. Frequency Domain Norm Blind Equalization Method

The principle of the frequency domain constant mode blind equalization method (FCMA) is shown in FIG. 4 . According to Figure 4, the signal before the fast Fourier transform FFT is divided into blocks. The length of each block signal is equal to the equalizer weight vector length L, and the weight vector is updated once every L samples, and each update is composed of L error signal samples Point accumulation result control, which ensures the same convergence speed as the time-domain blind equalization method CMA, and at the same time, through the fast Fourier transform (FFT) and the overlap-save method (OSL), the circular convolution is used to replace the linearity of the time-domain signal Convolution greatly reduces the amount of computation.

The error function of FCMA is defined as

$\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; f</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; Z</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 4</span>}<span\; class="notranslate">\; )</span>$

为发射信号a(n)的二阶频域统计模值。In the formula, N represents the number of data blocks whose length is L,

is the second-order frequency-domain statistical modulus of the transmitted signal a(n).

From the weight vector update formula of CMA, the weight vector update formula of FCMA is obtained as

F(N+1)=F(N)+μE(N)Z(N)Y ^* (N) (5)

In the formula, Z(N) is the output signal of the frequency domain equalizer, which is the fast Fourier transform of z(n); F(N) is the weight vector of the frequency domain equalizer, which is the fast Fourier transform of f(n) ; Y ^* (N) is the conjugate of the input signal Y(N) of the frequency domain equalizer, and Y(N) is the fast Fourier transform of y(n).

D. Modulo value transform frequency domain multi-mode blind equalization method

The modulus transformation is introduced into the frequency-domain normal-mode blind equalization method, and the modulus-value transformation frequency-domain multi-mode blind equalization method (MTFMMA) is obtained, as shown in Fig. 5 . The cost function of this method is

${\mathrm{<span\; class="notranslate">\; J</span>}}_{\mathrm{<span\; class="notranslate">\; MTFMMA</span>}}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; \{</span>{<span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; rtF</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; rt</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; ]</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; E.</span>}<span\; class="notranslate">\; \{</span><span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; itF</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}{<span\; class="notranslate">\; ]</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; \}</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

分别为发射信号a(n)的实部a_r(n)与虚部a_i(n)模值变换后的二阶统计模值的傅里叶变换。MTFMMA的权向量更新公式为In the formula, J _MTFMMA is the cost function of MTFMMA, Z _rt (N) and Z _it (N) are the real part and imaginary part of the frequency domain equalizer output signal Z _t (N) after the modulus transformation, respectively, and

are respectively the Fourier transform of the second-order statistical modulus after modulus transformation of the real part a _r (n) and the imaginary part a _i (n) of the transmitted signal a(n). The weight vector update formula of MTFMMA is

$f_r(N+1)=f_r\left(N\right)+\mathrm{\&mu;}\left(\right|R_{\mathrm{rtF}}^2-Z_{\mathrm{rt}}\left(N\right){|}^2)\left|Z_{\mathrm{rt}}\left(N\right)\right|\mathrm{sign}\left[Z_r\left(N\right)\right]Y_r^{*}\left(N\right)$ (7)

${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&mu;</span>}<span\; class="notranslate">\; (</span><span\; class="notranslate">\; |</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; itF</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; Z</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; Z</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>\mathrm{<span\; class="notranslate">\; sign</span>}<span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; Z</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>{\mathrm{<span\; class="notranslate">\; Y</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{<span\; class="notranslate">\; *</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span>$

与

分别为频域均衡器输入信号Y(N)的实部与虚部的共轭，分别为

和

的快速傅里叶变换；Z_r(N)和Z_i(N)分别表示频域均衡器输出Z(N)的实部和虚部，分别为z_r(n)和z_i(n)的快速傅里叶变换；F_r(N)和F_i(N)分别为频域均衡器权向量F(N)的实部和虚部，分别为f_r(n)和f_i(n)的快速傅里叶变换。In the formula,

and

are the conjugates of the real part and the imaginary part of the frequency domain equalizer input signal Y(N), respectively, and are

and

Fast Fourier transform of ; Z _r (N) and _Zi (N) represent the real part and imaginary part of the frequency domain equalizer output Z(N), respectively, and are z _r (n) and z _i (n) Fast Fourier transform; F _r (N) and F _i (N) are the real and imaginary parts of the frequency-domain equalizer weight vector F(N), respectively, and are the f _r (n) and f _i (n) Fast Fourier transform.

E. Fractional low-order statistic modulus transformation frequency-domain multi-mode blind equalization method

In α-stable distributed noise, using the fractional low-order statistics of the error function, the fractional low-order statistics modulus transform frequency-domain blind multi-mode equalization method (FLOSMTFMMA) will be obtained. The principle of this method is shown in Figure 6. The cost function of this method is

J _FLOSMTFMMA =E{|E _rt (N)| ² +|E _it (N)| ² } (8)

${\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; rt</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; rtF</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; Z</span>}}_{\mathrm{<span\; class="notranslate">\; rt</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; p</span>}}<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; itF</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; Z</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; p</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 9</span>}<span\; class="notranslate">\; )</span>$

和虚部p阶统计模值

的快速傅里叶变换，下同。由最速下降法，得分数低阶统计量模值变换频域多模盲均衡方法(FLOSMTFMMA)的权向量更新公式为In the formula, J _FLOSMTFMMA is the cost function of FLOSMTFMMA; p is a positive number greater than zero and less than 1, E _rt (N) and E _it (N) are the real part e of the time-domain error function e _t (n) of the analog transformation Fourier transform of _rt (n) and imaginary part e _it (n); and are the p-order statistical modulus of the real part of the transmitted signal a(n), respectively

and p-order statistical modulus of the imaginary part

The fast Fourier transform of , the same below. According to the steepest descent method, the weight vector update formula of the score low-order statistic modulus transformation frequency-domain multi-mode blind equalization method (FLOSMTFMMA) is

${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&mu;</span>}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; rt</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>$

$\&Center\; Dot;\mathrm{sign}\left({\mathrm{E.}}_{\mathrm{rt}}\left(N\right)\right){\left|Z_{\mathrm{rt}}\left(N\right)\right|}^{p-1}\mathrm{sign}\left(Z_{\mathrm{rt}}\left(N\right)\right)Y_r^{*}\left(N\right)$ (10)

${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&mu;</span>}<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>$

$<span\; class="notranslate">\; \&Center\; Dot;</span>\mathrm{<span\; class="notranslate">\; sign</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; Z</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; sign</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; Z</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; Y</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{<span\; class="notranslate">\; *</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span>$

Fractional low-order statistics modulus transform frequency domain multi-mode blind equalization method, by combining fractional low-order statistics with MTFMMA, FLOSMTFMMA can also play a better performance under the condition of α-stable distributed noise.

F. A fractional low-order statistic modulus transform wavelet frequency domain multi-mode blind equalization method of the present invention

Before the signal enters the FLOSMTFMMA equalizer, after the orthogonal wavelet transform, a fractional low-order statistic modulus transform wavelet frequency-domain multi-mode blind equalization method WT-FLOSMTFMMA is invented. Its principle, as shown in Figure 1. In Figure 1, a(n) is the transmitted signal, c(n) is the impulse response vector of the channel, and c(n)=[c(n),…,c(n-M+1)] ^T (superscript T represents the transpose, M represents the length of the channel), b(n) is the signal vector output through the channel, w(n) is the α stable distribution noise, y(n) is the signal vector after passing through the channel and mixed with noise, v _r (n) and v _i (n) are the signal vectors corresponding to the real part y _r (n) and imaginary part y _i (n) of y(n) after orthogonal wavelet transformation, R _r (N) and R _i (N) are the signal vectors of v _r (n) and v _i (n) after fast Fourier transform respectively, Z _r (N) and Z _i (N) are the output signals of the frequency domain equalizer, z _r ( n) and z _i (n) are the inverse fast Fourier transformed signals of Z _r (N) and _Zi (N) respectively, and z _rt (n) and z _it (n) are the signals after modulus transformation The real and imaginary parts of the signal vector z _t (n), e _rt (n) and e _it (n) are the real and imaginary parts of the time-domain error function after modulus transformation, E _rt (N) and E _it (N) is the real part and imaginary part of the error function E(N) after e _rt (n) and e _it (n) are Fourier transformed, and z(n) is the time domain equalization of the final combined output output signal. It can be seen from Figure 1

y(n)=c ^T (n)a(n)+w(n) (11)

v _r (n)=Qy _r (n), v _i (n)=Qy _i (n) (12)

${\mathrm{<span\; class="notranslate">\; e</span>}}_{\mathrm{<span\; class="notranslate">\; rt</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; rt</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; rt</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; p</span>}}<span\; class="notranslate">\; ,</span>$ ${\mathrm{<span\; class="notranslate">\; e</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}^{<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span>{<span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; z</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; p</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 13</span>}<span\; class="notranslate">\; )</span>$

In the formula, Q is the orthogonal wavelet transform matrix, and the output of the wavelet frequency domain equalizer is

Z _r (N)=R _r (N)F _r (N), Z _i (N)=R _i (N)F _i (N) (14)

At this time, the weight vector iteration formula of WT-FLOSMTFMMA is

${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&mu;</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; rt</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>$

$\&Center\; Dot;\mathrm{sign}\left({\mathrm{E.}}_{\mathrm{rt}}\left(N\right)\right){\left|Z_{\mathrm{rt}}\left(N\right)\right|}^{p-1}\mathrm{sign}\left(Z_{\mathrm{rt}}\left(N\right)\right)R_r^{*}\left(N\right)$ (15)

${\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; \&mu;</span>}{\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>$

$<span\; class="notranslate">\; \&CenterDot;</span>\mathrm{<span\; class="notranslate">\; sign</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; E.</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; Z</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; p</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}\mathrm{<span\; class="notranslate">\; sign</span>}<span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; Z</span>}}_{\mathrm{<span\; class="notranslate">\; it</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; )</span>{\mathrm{<span\; class="notranslate">\; R</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}^{<span\; class="notranslate">\; *</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; N</span>}<span\; class="notranslate">\; )</span>$

和

表示频域均衡器输入信号R(N)的实部R_r(N)与虚部R_i(N)的共轭，

为

的快速傅里叶变换，且

的获取公式为In the formula,

and

Represents the conjugate of the real part R _r (N) and the imaginary part R _i (N) of the frequency domain equalizer input signal R(N),

for

The fast Fourier transform of , and

The acquisition formula is

${\stackrel{<span\; class="notranslate">\; ^</span>}{\mathrm{<span\; class="notranslate">\; R</span>}}}^{<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; diag</span>}<span\; class="notranslate">\; [</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 0</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span><span\; class="notranslate">\; \&CenterDot;</span><span\; class="notranslate">\; \&Center\; Dot;</span><span\; class="notranslate">\; \&Center\; Dot;</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1,0</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ,</span><span\; class="notranslate">\; \&Center\; Dot;</span><span\; class="notranslate">\; \&Center\; Dot;</span><span\; class="notranslate">\; \&Center\; Dot;</span><span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; k</span>}}_{\mathrm{<span\; class="notranslate">\; J</span>}}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; ]</span>$

分别表示对u_j，k(n)与

的平均功率估计，可由下式递推得到Among them, diag[ ] represents a diagonal matrix, and

Respectively for u _{j, k} (n) and

The average power estimate of can be recursively obtained by the following formula

${\mathrm{\&sigma;}}_{j,m}^2(\mathrm{no}+1)={\mathrm{\&beta;}}_{\mathrm{\&sigma;}}{\mathrm{\&sigma;}}_{j,m}^2\left(\mathrm{no}\right)+(1-{\mathrm{\&beta;}}_{\mathrm{\&sigma;}})|u_{j,m}\left(\mathrm{no}\right){|}^2$ (16)

${\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}_{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}_{\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; +</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&beta;</span>}}_{\mathrm{<span\; class="notranslate">\; \&sigma;</span>}}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; |</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; J</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; m</span>}}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span>{<span\; class="notranslate">\; |</span>}^{\mathrm{<span\; class="notranslate">\; 2</span>}}$

In the formula, u _{j, m} (n) is the wavelet transform coefficient with scale parameter j and translation parameter m, s _{J, m} (n) is the scale transformation coefficient with scale parameter J and translation parameter m, and j is the scale Parameters, m∈Z is the translation parameter, J is the maximum scale of wavelet decomposition, k is the translation parameter of the corresponding wavelet function under the scale parameter j, k _J is the maximum translation of the wavelet function under the scale J, β _σ is the smoothing factor, and 0 <β _σ <1.

After the orthogonal wavelet transform, the autocorrelation matrix of the signal is closer to the diagonal matrix. At this time, the energy of the signal is mainly concentrated near the diagonal, that is, the correlation of the signal becomes smaller. Therefore, the WT-FLOSMTFMMA method of the present invention has the characteristics of fast convergence speed and small mean square error, and its performance is improved.

Implementation example

In order to verify the effectiveness of the WT-FLOSMTFMMA method of the present invention, experiments were carried out with FLOSFCMA and FLOSMTFMMA methods as comparison objects. In the experiment, the step sizes of FLOSFCMA, FLOSMTFMMA and WT-FLOSMTFMMA are 0.0002, 0.0002, and 0.004 respectively, and the Db2 wavelet is used for decomposition. The decomposition level is two layers. The initial value of wavelet power is 4, and the channel impulse response vector c=[0.9656,- 0.0906, 0.0578, 0.2368], the number of equalizer taps is 16, both FLOSMTFMMA and WT-FLOSMTFMMA use the 12th tap to initialize to 1, FLOSFCMA uses the 8th tap to initialize to 1, the generalized signal-noise-ratio (generalized signal-noise-ratio , GSNR) is 25dB. In α-stable distributed noise, when p=0.8499 in WT-FLOSMTFMMA and FLOSMTFMMA, the 700 Monte Carlo simulation results are shown in Figure 7; when p=0.63 in WT-FLOSMTFMMA and p=0.6 in FLOSMTFMMA , 700 Monte Carlo simulation results are shown in Figure 8.

Figure 7(a) shows that the convergence speed of the inventive method WT-FLOSMTFMMA is about 3000 steps faster than FLOSFCMA and FLOSMTFMMA, and the steady-state error of the inventive method WT-FLOSMTFMMA is about 2dB smaller than that of FLOSFCMA. Fig. 8(a) shows that the convergence speed of the method WT-FLOSMTFMMA of the present invention is about 2500 steps faster than that of FLOSMTFMMA.

Figure 7(b)~(d) and Figure 8(b)~(d) show that the eye diagram opening effect of the method WT-FLOSMTFMMA and FLOSMTFMMA of the present invention is the same, but both are clearer than the eye diagram opening effect of the method FLOSFCMA.

It can be seen from Fig. 7 and Fig. 8 that in the α-stable distributed noise, the adaptability of the method WT-FLOSMTFMMA of the present invention is the strongest.

Claims (2)

1. a fractional lower-order statistics mould value transform wavelet frequency domain multimode blind balance method, is characterized in that, comprises the steps:

Steps A, a (n) that will transmit obtains channel output vector b (n) through impulse response channel c (n), and wherein n is time series;

Step B adopts α to stablize partition noise w (n) and the described channel output vector of steps A b (n) obtains the front equalizer input signal y (n) of orthogonal wavelet transformation: y (n)=w (n)+b (n);

Step C gets its real part y to the described equalizer input signal y of step B (n) _r(n) and imaginary part y _i(n), then to real part y _r(n) and imaginary part y _i(n) carry out respectively orthogonal wavelet transformation, through the signal after orthogonal wavelet transformation be

v _r(n)=Qy _r(n)，v _i(n)=Qy _i(n)

In formula, Q is the orthogonal wavelet transformation matrix, v _r(n) and v _i(n) be respectively the real part y of time-domain equalizer input signal y (n) _r(n) and imaginary part y _i(n) through the signal component after orthogonal wavelet transformation, the real part Z of frequency-domain equalizer output Z (N) _r(N) and imaginary part Z _i(N) be respectively

Z _r(N)=R _r(N)F _r(N)，Z _i(N)=R _i(N)F _i(N)

In formula, N represents that length is the piece number of the data block of L, and L is equalizer weight vector length, F _r(N) and F _i(N) be respectively real part and the imaginary part of frequency-domain equalizer weight vector F (N), R _r(N) and R _i(N) be respectively v _r(n) and v _i(n) through frequency domain real part and imaginary part after fast Fourier transform;

Step D is to the real part Z of step C frequency-domain equalizer output signal Z (N) _r(N) and imaginary part Z _i(N) obtain respectively the real part z of time-domain equalizer output signal z (n) as Fourier inversion _r(n) and imaginary part z _i(n).

2. a kind of fractional lower-order statistics mould value transform wavelet frequency domain multimode blind balance method according to claim 1, is characterized in that, in described step C, the calculation procedure of frequency-domain equalizer weight vector F (N) is as follows:

Step C-1 calculates mould value transform time domain error function e _t(n) real part e _rt(n) with imaginary part e _it(n), computing formula is as follows:

$e_{\mathrm{rt}}\left(n\right)=R_{\mathrm{rt}}^{\left(p\right)}-|z_{\mathrm{rt}}\left(n\right){|}^p,$ $e_{\mathrm{it}}\left(n\right)=R_{\mathrm{it}}^{\left(p\right)}-|z_{\mathrm{it}}\left(n\right){|}^p$

$R_{\mathrm{rt}}^{\left(p\right)}=\frac{E\left\{\right|a_{\mathrm{rt}}\left(n\right){|}^{2p}\}}{E\left\{\right|a_{\mathrm{rt}}\left(n\right){|}^p\}},$ $R_{\mathrm{it}}^{\left(p\right)}=\frac{E\left\{\right|a_{\mathrm{it}}\left(n\right){|}^{2p}\}}{E\left\{\right|a_{\mathrm{it}}\left(n\right){|}^p\}}$

In formula, || be modulo operation, p be greater than zero less than 1 positive number, E{} represents to ask mathematic expectaion, a _rt(n) and a _it(n) be respectively the real part a of a that transmits (n) _r(n) with imaginary part a _i(n) pass through respectively signal component after the mould value transform,

With

Be respectively a _rt(n) and a _it(n) p rank statistics mould value, z _rt(n) and z _it(n) be respectively the real part z of time-domain equalizer output signal z (n) _r(n) with imaginary part z _i(n) through real part and imaginary part after the mould value transform; Step C-2 is to mould value transform time domain error function e _t(n) real part e _rt(n) with imaginary part e _it(n) make Fourier transform after, obtain mould value transform error of frequency domain function E _t(N) real part E _rt(N) with imaginary part E _it(N);

Step C-3 calculates frequency-domain equalizer weight vector F (N), and the formula of its iterative process is:

$F_r(N+1)=F_r\left(N\right)+\mathrm{\&mu;}{\hat{R}}^{-1}\left(N\right)\left|E_{\mathrm{rt}}\left(N\right)\right|$

$\&CenterDot;\mathrm{sign}\left(E_{\mathrm{rt}}\left(N\right)\right)|Z_{\mathrm{rt}}\left(N\right){|}^{p-1}\mathrm{sign}\left(Z_{\mathrm{rt}}\left(N\right)\right)R_r^{*}\left(N\right)$

$F_i(N+1)=F_i\left(N\right)+\mathrm{\&mu;}{\hat{R}}^{-1}\left(N\right)\left|E_{\mathrm{it}}\left(N\right)\right|$

$\&CenterDot;\mathrm{sign}\left(E_{\mathrm{it}}\left(N\right)\right)|Z_{\mathrm{it}}\left(N\right){|}^{p-1}\mathrm{sign}\left(Z_{\mathrm{it}}\left(N\right)\right)R_i^{*}\left(N\right)$

In formula, μ is iteration step length, and symbolic operation, Z are got in sign () expression _rt(N) and Z _it(N) be respectively through the frequency-domain equalizer output signal Z after the mould value transform _t(N) real part and imaginary part; With The real part R of expression frequency-domain equalizer input signal R (N) _r(N) with imaginary part R _i(N) conjugation;

For

Fast Fourier transform, and

The formula that obtains be

${\hat{R}}^{-1}\left(n\right)=\mathrm{diag}[{\mathrm{\&sigma;}}_{j,0}^2\left(n\right),{\mathrm{\&sigma;}}_{j,1}^2\left(n\right),\&CenterDot;\&CenterDot;\&CenterDot;,{\mathrm{\&sigma;}}_{j,k}^2\left(n\right),{\mathrm{\&sigma;}}_{J+\mathrm{1,0}}^2\left(n\right),\&CenterDot;\&CenterDot;\&CenterDot;,{\mathrm{\&sigma;}}_{J+1,k_J-1}^2\left(n\right)]$

Wherein, diag[] the expression diagonal matrix,

With

Represent u respectively _{J, k}(n) with

Average power estimate, can be obtained by the following formula recursion:

${\mathrm{\&sigma;}}_{j,m}^2(n+1)={\mathrm{\&beta;}}_{\mathrm{\&sigma;}}{\mathrm{\&sigma;}}_{j,m}^2\left(n\right)+(1-{\mathrm{\&beta;}}_{\mathrm{\&sigma;}})|u_{j,m}\left(n\right){|}^2$

${\mathrm{\&sigma;}}_{J+1,m}^2(n+1)={\mathrm{\&beta;}}_{\mathrm{\&sigma;}}{\mathrm{\&sigma;}}_{J+1,m}^2\left(n\right)+(1-{\mathrm{\&beta;}}_{\mathrm{\&sigma;}})|s_{J,m}\left(n\right){|}^2$

In formula, u _{J, m}(n) be that scale parameter is that j, translation parameters are the wavelet conversion coefficient of m, s _{J, m}(n) for scale parameter is that J, translation parameters are the change of scale coefficient of m, J is the out to out of wavelet decomposition, and k is the translation parameters of corresponding wavelet function under scale parameter j, k _JThe expression out to out is the maximal translation of wavelet function under J, β _σSmoothing factor, and 0＜β _σ＜1.

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